We study the robustness of Bayesian persuasion to uncertainty about the receiver's preferences. We analyze two conceptually distinct notions: continuity, in which only the modeler lacks precise knowledge, but where the model's predictions are nonetheless accurate; and robustness, in which the sender also lacks precise knowledge, but where the outcome is insensitive to this ignorance. We model preference uncertainty as infinitesimally small, non-probabilistic (Knightian) uncertainty, and the sender's behavior as either minimizing the regret or maximizing the minimum utility. We show that continuity holds if and only if robustness holds, and that both notions are generic. Thus, while some instances of Bayesian persuasion are fragile, typical instances are both continuous and robust with respect to a small amount of ignorance.

We study games with incomplete information and characterize when a feasible outcome is Pareto efficient. Outcomes with excessive randomization are inefficient: generically, the total number of action profiles across states must be strictly less than the sum of the number of players and the number of states. We consider three applications. A cheap talk outcome is efficient only if pure; with state-independent sender payoffs, it is efficient if and only if the sender's most preferred action is induced with certainty. In natural settings, Bayesian persuasion outcomes are inefficient across many priors. Finally, ranking-based allocation mechanisms are inefficient under mild conditions.

The canonical Bayesian persuasion setting studies a model where an informed agent, the Sender, can partially share his information with an uninformed agent, the Receiver. The Receiver's utility is a function of the state of nature and the Receiver's action while the Sender's is only a function of the Receiver's action. The classical results characterize the Sender's optimal information disclosure policy whenever the state space is finite. In this paper we study the same setting where the state space is an interval on the real line. We introduce the class of bi-pooling policies and the induced distribution over posteriors which we refer to as bi-pooling distributions. We show that this class of distributions characterizes the set of optimal distributions in the aforementioned setting. Every persuasion problem admits an optimal bi-pooling distribution as a solution. Conversely, for every bi-pooling distribution there exists a persuasion problem in which the given distribution is the unique optimal one. We leverage this result to study the structure of the price function (see [1]) in this setting and to identify optimal information disclosure policies. The full paper can be accessed at https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3511516.

Society uses the following mechanism to decide on the supply of an experience good. Each agent can choose whether or not to contribute to the good. Contributions are collected, and the good is supplied whenever total contributions exceed a threshold. We study the case where the good is excludable, agents have a common value, and each agent receives a private signal about the common value. We study how such collective decisions perform in terms of information aggregation, social efficiency, and market traction.

We study the robustness of cheap-talk equilibria to infinitesimal private information of the receiver in a model with a binary state-space and state-independent sender-preferences. We show that the sender-optimal equilibrium is robust if and only if this equilibrium either reveals no information to the receiver or fully reveals one of the states with positive probability. We then characterize the actions that can be played with positive probability in any robust equilibrium. Finally, we fully characterize the optimal sender-utility under binary receiver's private information, and provide bounds for the optimal sender-utility under general private information.

The classic herding model examines the asymptotic behavior of agents who observe their predecessors' actions as well as a private signal from an exogenous information structure. In this paper, we introduce a self-interested sender into the model and study her prob-lem of designing this information structure. If agents cannot observe each other, the model reduces to Bayesian persuasion. However, when agents observe predecessors' actions, they may learn from them, potentially harming the sender. We identify necessary and suf-ficient conditions under which the sender can nevertheless obtain the same utility as when the agents are unable to observe each other. (JEL D82, D83, D91)

In this paper we revisit the basic variant of the classical secretary problem. We propose a new approach in which we separate between an agent that evaluates the secretary performance and one that has to make the hiring decision. The evaluating agent (the sender) signals the quality of the candidate to the hiring agent (the receiver) who must make a decision. Whenever the two agents' interests are not fully aligned, this induces an information transmission (signaling) challenge for the sender. We study the sender's optimization problem subject to persuasiveness constraints of the receiver for several variants of the problem. Our results quantify the loss in performance for the sender due to online arrival. We provide optimal and near-optimal persuasive mechanisms that recover at least a constant fraction of a natural utility benchmark for the sender. The separation of evaluation and decision making can have a substantial impact on the approximation results. While in some scenarios, techniques and results closely mirror the conditions in the standard secretary problem, we also explore conditions that lead to very different characteristics.

A data curator would like to collect data from privacy-aware agents. The collected data will be used for the benefit of all agents. Can the curator incentivize the agents to share their data truthfully? Can he guarantee that truthful sharing will be the unique equilibrium? Can he provide some stability guarantees on such equilibrium? We study necessary and sufficient conditions for these questions to be answered positively and complement these results with corresponding data collection protocols for the curator. Our results account for a broad interpretation of the notion of privacy awareness. The full version of this paper is available at https://arxiv.org/abs/2207.06929 .

The Bayesian persuasion model studies communication between an informed sender and a receiver with a payoff-relevant action, emphasizing the ability of a sender to extract maximal surplus from his informational advantage. In this paper, we study a setting with multiple senders in which the receiver is restricted to choosing, at the interim stage, one sender with whom to interact. Our main result is that whenever senders are uncertain about each other's prefer-ences and, in particular, cannot dismiss with certainty the possibility that others are aligned with the receiver, the receiver receives all the informational surplus in all equilibria. (JEL C72, D82, D83)

The Bayesian persuasion paradigm of strategic communication models interaction between a privately-informed agent, called the sender, and an ignorant but rational agent, called the receiver. The goal is typically to design a (near-)optimal communication (or signaling) scheme for the sender. It enables the sender to disclose information to the receiver in a way as to incentivize her to take an action that is preferred by the sender. Finding the optimal signaling scheme is known to be computationally difficult in general. This hardness is further exacerbated when there is also a constraint on the size of the message space, leading to NP-hardness of approximating the optimal sender utility within any constant factor. In this paper, we show that in several natural and prominent cases the optimization problem is tractable even when the message space is limited. In particular, we study signaling under a symmetry or an independence assumption on the distribution of utility values for the actions. For symmetric distributions, we provide a novel characterization of the optimal signaling scheme. It results in a polynomial-time algorithm to compute an optimal scheme for many compactly represented symmetric distributions. In the independent case, we design a constant-factor approximation algorithm, which stands in marked contrast to the hardness of approximation in the general case.

We study the implications of endogenous pricing for learning and welfare in the classic herding model . When prices are determined exogenously, it is known that learning occurs if and only if signals are unbounded. By contrast, we show that learning can occur when signals are bounded as long as non-conformism among consumers is scarce. More formally, learning happens if and only if signals exhibit the vanishing likelihood property introduced bellow. We discuss the implications of our results for potential market failure in the context of Schumpeterian growth with uncertainty over the value of innovations.

It is well understood that the structure of a social network is critical to whether or not agents can aggregate information correctly. In this paper, we study social networks that support information aggregation when rational agents act sequentially and irrevocably. Whether or not information is aggregated depends, inter alia, on the order in which agents decide. Thus, to decouple the order and the topology, our model studies a random arrival order.Unlike the case of a fixed arrival order, in our model, the decision of an agent is unlikely to be affected by those who are far from him in the network. This observation allows us to identify a local learning requirement, a natural condition on the agent's neighborhood that guarantees that this agent makes the correct decision (with high probability) no matter how well other agents perform. Roughly speaking, the agent should belong to a multitude of mutually exclusive social circles.We illustrate the power of the local learning requirement by constructing a family of social networks that guarantee information aggregation despite that no agent is a social hub (in other words, there are no opinion leaders). Although the common wisdom of the social learning literature suggests that information aggregation is very fragile, another application of the local learning requirement demonstrates the existence of networks where learning prevails even if a substantial fraction of the agents are not involved in the learning process. On a technical level, the networks we construct rely on the theory of expander graphs, i.e., highly connected sparse graphs with a wide range of applications from pure mathematics to error-correcting codes.

Differential privacy is commonly used in the computer science literature as a mathematical definition of privacy for the purpose of quantifying and bounding privacy loss. It induces a preference order over the set of privacy-jeopardizing mechanisms which, in turn, adhere to some properties of this order. We show that a set of five such properties uniquely captures the ordinal implications of prioritizing the alternatives in agreement with differential privacy. The model can also be applied to evaluate the appropriateness of differential privacy in different settings.

In various situations, decision makers face experts that may provide conflicting advice. This advice may be in the form of probabilistic forecasts over critical future events. We consider a setting where the two forecasters provide their advice repeatedly and ask whether the decision maker can learn to compare and rank the two forecasters based on past performance. We take an axiomatic approach and propose three natural axioms that a comparison test should comply with. We propose a test that complies with our axioms. Perhaps, not surprisingly, this test is closely related to the likelihood ratio of the two forecasts over the realized sequence of events. More surprisingly, this test is essentially unique. Furthermore, using results on the rate of convergence of supermartingales, we show that whenever the two experts\textquoteright{} advice are sufficiently distinct, the proposed test will detect the informed expert in any desired degree of precision in some fixed finite time.

Privacy, in the sense of control over access to one’s personal information, is a central concern in the context of online decision making, both in general and in relation to online platforms in particular. For at least some agents, a belief that one online platform jeopardizes users’ privacy more than another may tip the scales in favor of the latter. Thus, understanding how privacy considerations come into play is central for any economic or social analysis. To this end, we study how agents rank online platforms (or mechanisms, as we call them) from a privacy perspective. We propose a very simple model of privacy-jeopardizing mechanisms, along with a normative methodology for understanding how these mechanisms are ranked. Similarly to classic work in decision theory, we postulate several axioms that we believe a privacy order should satisfy, and then characterize the set of orders that comply with these axioms. These orders turn out to be related to the notion of f-divergence from information theory, one example of which is KL divergence. We test the usefulness of our theoretical result by using it to rank clustering models based on data provided by the Recommendation Team at Microsoft Research.

We study an information-structure design problem (i.e., a Bayesian persuasion problem) in an online scenario. Inspired by the classic gambler's problem, consider a set of candidates who arrive sequentially and are evaluated by one agent (the sender). This agent learns the value from hiring the candidate to herself as well as the value to another agent, the receiver. The sender provides a signal to the receiver who, in turn, makes an irrevocable decision on whether or not to hire the candidate. A-priori, for each agent the distribution of valuation is independent across candidates but may not be identical. We design good online signaling schemes for the sender. To assess the performance, we compare the expected utility to that of an optimal offline scheme by a prophet sender who knows all candidate realizations in advance. We show an optimal prophet inequality for online Bayesian persuasion, with a 1/2-approximation when the instance satisfies a "satisfactory-status-quo" assumption. Without this assumption, there are instances without any finite approximation factor. We extend the results to combinatorial domains and obtain prophet inequalities for matching with multiple hires and multiple receivers.

Consider a setting where many individuals forecast the (unknown) state of nature based on signals they receive independently. We refer to the joint distribution over the states and signals as an “information structure.” An information structure is deemed identifiable if the distribution of forecasts is sufficient to determine the state of nature, even without knowing the underlying information structure. We characterize the set of identifiable information structures and propose a scheme that uniquely identifies the state of nature for the finite case.

We consider social learning where agents can only observe part of the population (modeled as neighbors on an undirected graph), face many decision problems, and arrival order of the agents is unknown. The central question we pose is whether there is a natural observability graph that prevents the information cascade phenomenon. We introduce the ‘celebrities graph’ and prove that indeed it allows for proper information aggregation in large populations even when the order at which agents decide is random and even when different issues are decided in different orders.

In the classical secretary problem, multiple secretaries arrive one at a time to compete for a single position, and the goal is to choose the best secretary to the job while knowing the candidate's quality only with respect to the preceding candidates. In this paper we define and study a new variant of the secretary problem, in which there are multiple jobs. The applicants are ranked relatively upon arrival as usual, and, in addition, we assume that the jobs are also ranked. The main conceptual novelty in our model is that we evaluate a matching using the notion of blocking pairs from Gale and Shapley's stable matching theory. Specifically, our goal is to maximize the number of matched jobs (or applicants) that do not take part in a blocking pair. We study the cases where applicants arrive randomly or in adversarial order, and provide upper and lower bounds on the quality of the possible assignment assuming all jobs and applicants are totally ordered. Among other results, we show that when arrival is uniformly random, a constant fraction of the jobs can be satisfied in expectation, or a constant fraction of the applicants, but not a constant fraction of the matched pairs.

A policy maker faces a sequence of unknown outcomes. At each stage two (self-proclaimed) experts provide probabilistic forecasts on the outcome in the next stage. A comparison test is a protocol for the policy maker to (eventually) decide which of the two experts is better informed. The protocol takes as input the sequence of pairs of forecasts and actual outcomes and (weakly) ranks the two experts. We focus on anonymous and non-counterfactual comparison tests and propose two natural properties to which such a comparison test must adhere. We show that these determine the test in an essentially unique way. The resulting test is a function of the derivative of the induced pair of measures at the realized outcomes.